We have built a microwave Fabry-Perot resonator made of diamond-machined copper mirrors coated with superconducting niobium. Its damping time (Tc = 130 ms at 51 GHz and 0.8 K) corresponds to a finesse of 4.6 x 109, the highest ever reached for a Fabry-Perot in any frequency range. This result opens novel perspectives for quantum information, decoherence and non-locality studies.